Main

Expansion Points

Up until now, Taylor series expansions have all been at . The Taylor series at gives a good approximation to the function near 0. But what if we want a good approximation to the function near a different point ? That is the topic of this module.

Expansion points

A function has a Taylor series expansion about any point provided that and all its derivatives exist at . The definition of the Taylor series for about is

We say this is a series in . A different way to view this series is by making the change of variables . After cancellation, this yields

Taylor polynomial for approximation

Recall that the first few terms of the Taylor series for about gives a polynomial (the Taylor polynomial) which is a good approximation for near 0. Similarly, the Taylor polynomial for about gives a polynomial which is a good approximation of near . Note, however, that as the input gets further away from the expansion point , the approximation gets worse.

Example Find the Taylor series for about .

Computing the derivatives, and evaluating at , one finds

And all the subsequent derivatives are 0. So from the definition, one finds that

This appears to be different than the polynomial with which we began. If one multiplies out this polynomial and collects like terms, however, the result is the original polynomial. This should not be surprising, since the best polynomial approximation to a polynomial is the polynomial itself, even factored into a slightly different form.

Example Compute the Taylor series expansion for about .

Begin by computing the first few derivatives and evaluating at :

The pattern that emerges is . To see that the pattern holds, check that

as desired. So by induction, the pattern holds. It follows that for . Plugging in to the formula, one finds that

Note that with the change of variables (and hence , we find that

which is the same series we found earlier for .

Note that the Taylor polynomial is only a good approximation to the function on the domain of convergence. For functions whose domain of convergence is the entire number line, this is not a concern. But for functions such as , the Taylor polynomials will only be a good approximation within the domain of convergence, which is . Outside of that domain, the Taylor polynomials diverge wildly from , as shown here:

Even within a function's domain of convergence, a Taylor polynomial's approximation gets worse as the input gets further away from . One way to improve an approximation is to include more and more terms of the Taylor series in the Taylor polynomial. However, this involves computing more and more derivatives. Another way to improve the approximation for is to choose an expansion point which is close to .

Example Use the Taylor polynomial of degree 2 for about to approximate . Then repeat the process about and compare the results.

Using the definition, one finds

Thus, the Taylor polynomial about is

And the corresponding approximation is

which is obviously quite far off the mark. On the other hand, the Taylor polynomial about is

And the corresponding approximation is

which is quite a good approximation of .

Caveat for compositions

When computing the Taylor expansion for the composition about , one must be careful of expansion points. In particular, one cannot simply take the series for at and plug it into the series for at .

Example Consider the expansion for about . Although , and , one will run into trouble trying to write

The trouble is that collecting like terms requires adding up infinitely many things. For instance, the constant term above is . The reason this is a problem is that Taylor series are supposed to give a good polynomial approximation of a function without requiring too much computation or information about the function.

Remember that is a good approximation when is near 0. However, when is near 0, is near 1. So plugging the series for into the series for does not give a good approximation.

To avoid this problem when computing the Taylor series for the composition at , one should plug the Taylor expansion of about into the expansion of about . In the above example, the expansion of about is

so


EXERCISES:

  • Without using a calculator, find a decimal approximation to by Taylor-expanding about and using the zero-th and first order terms.
  • Without using a calculator, find a decimal approximation to using linear approximation. How close was your answer to truth?
  • Without using a calculator, find a decimal approximation to [in radians!] using linear approximation. How close was your answer to truth? (Hint: ...)
  • Taylor expand about and compute all the terms. Does what you get make sense?
  • Use completing the square and the geometric series to get the Taylor expansion about of
  • Approximate using the zeroth and first order terms of the Taylor series.